Current developments in bioequivalent technology have led to the creation of excellent models that mimic the structure and function of human organs. These models are based on the original tissues and organs of the human body, but they lack the complex interaction with the extensive network of vasculature, and this is a major challenge for these models. A functional vasculature is essential for oxygen, nutrient, and waste exchange. It is also responsible for inductive biochemical exchange, and provides a structural pattern for organ growth. In vitro systems, containing no perfusable vessels, suffer from the quick formation of a necrotic core of organoids, and further development does not occur due to increased metabolic demands. Another key limitation of 3D-based techniques is the absence of accurate architectural structures and large-scale tissue sizes. Recently, new 3D bioprinting methods have been developed for organoids and spheroids as living building blocks. These methods aim to address some of the challenges associated with 3D technologies. In this review, we discuss recent strategies for vascularization via organoids and spheroids, which are used as structural units in bioprinting to recreate natural organs and tissues with ever-increasing accuracy in structure and function.